Steam Deck natively recognizes the PS3 controller. Simply plug it in via USB, wait for Steam OS to recognize it, then unplug USB cable and it will work over Bluetooth without needing
DualShock - Wikipedia
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Official Genuine Sony Playstation 3 - DualShock 3 Controller - Multiple Colours | eBay
Identifying a Fake PlayStation 3 Dual Shock 3 Controller | Teardown & Comparison | PS3 - YouTube
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DualShock - Wikipedia
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Official Original Sony Playstation Dual Shock 3 PS3 Controller Multiple Colours | eBay
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Official Sony PlayStation 3 PS3 Controller OEM Original Authentic TESTED | eBay
User blog:Himura Kensh/List of Ps3 controller variants | PlayStation Wiki | Fandom
For those looking for legitimate new, or pre-owned dual shock 3 controllers I found the following reference for making absolutely sure that what you get is original, and not fake. : r/PS3